Team:Freiburg Bioware/Modeling/Virus Production

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<h1>Model for Virus Production</h1>
<h1>Model for Virus Production</h1>
<br>
<br>
<h2>Reaction Scheme</h2>
<h2>Reaction Scheme</h2>
-
Reducing the complexity of virus production we divide the cell into three compartments: the <b>extracellular matrix</b> (all quantities with the index <i>ext</i>), the <b>cytoplasm</b> (<i>cyt</i>) and the <b>nucleus</b> (<i>nuc</i>). Four plasmids are transfected - the plasmid coding for the <b>helper proteins</b> (<i>helper</i>), the <b>gene of interest</b> (<i>goi</i>) and two types of plasmids coding for the <b>capsid proteins</b> (<i>capwt</i> [wild type], <i>capmod</i> [modified]).<br>
+
<p style="text-align: justify;">
-
The plasmids are transported into the nucleus where gene expression is initiated. Processed mRNA is transported into the cytoplasm and <b>proteins</b> (<i>phelper</i>, <i>pcapwt</i>, <i>pcapmod</i>) are produced. Containing a nuclear localization sequence proteins are relocated into the nucleus where capsid assembly occurs. The viral capsid is compose of 60 subunits of viral coat proteins. Titration of the two plasmids coding for the capsid proteins leads to virus surfaces with different ratios of wild type and modified capsid proteins.<br>
+
Reducing the complexity of virus production we divide the cell into
-
The gene of interest is replicated by cellular polymerases and <b>single stranded DNA</b> (<i>ssDNA</i>) is encapsidated into the preformed <b>capsids</b> (<i>capsid</i>) forming infectious <b>viral particles</b> (<i>V</i>).<br>
+
three compartments: the <b>extracellular matrix</b> (all quantities
-
Finally the recombinant viruses are released into the extracellular matrix and can be harvested for transduction.
+
with the index <i>ext</i>), the <b>cytoplasm</b> (<i>cyt</i>) and the
-
 
+
<b>nucleus</b> (<i>nuc</i>). Four plasmids are transfected - the
-
<br><br>
+
plasmid coding for the <b>helper proteins</b> (<i>helper</i>), the <b>gene
-
 
+
of interest</b> (<i>goi</i>) and two types of plasmids coding for the <b>capsid
-
 
+
proteins</b> (<i>capwt</i> [wild type], <i>capmod</i> [modified]).<br>
-
<table>
+
The plasmids are transported into the nucleus where gene expression is
-
<td>
+
initiated. Processed mRNA is transported into the cytoplasm and <b>proteins</b>
-
<center>
+
(<i>phelper</i>, <i>pcapwt</i>, <i>pcapmod</i>) are produced.
-
<img width="477" src="https://static.igem.org/mediawiki/2010/c/c9/Freiburg10_VirusProductionScheme01.png" alt="Reaction scheme for the virus production" />
+
Containing a nuclear localization sequence proteins are relocated into
-
</center>
+
the nucleus where capsid assembly occurs. The viral capsid is composed
 +
of 60 subunits of viral coat proteins. Titration of the two plasmids
 +
coding for the capsid proteins leads to virus surfaces with different
 +
ratios of wild type and modified capsid proteins.<br>
 +
The gene of interest is replicated by cellular polymerases and <b>single
 +
stranded DNA</b> (<i>ssDNA</i>) is encapsidated into the preformed <b>capsids</b>
 +
(<i>capsid</i>) forming infectious <b>viral particles</b> (<i>V</i>).<br>
 +
Finally the recombinant viruses are released into the extracellular
 +
matrix and can be harvested for transduction.
 +
</p>
 +
<br>
 +
<br>
 +
<table
 +
style="width: 905px; height: 808px; text-align: left; margin-left: auto; margin-right: auto;">
 +
<tbody>
 +
<tr>
 +
<td style="vertical-align: top;"><img
 +
alt="Reaction scheme for the virus production"
 +
src="https://static.igem.org/mediawiki/2010/c/c9/Freiburg10_VirusProductionScheme01.png"
 +
width="370"><br>
 +
<br>
 +
<br>
 +
<br>
 +
<div style="text-align: justify;"><span style="font-weight: bold;">Figure
 +
1:</span> Schematic overview of virus production: A production cell
 +
line is&nbsp; transfected with 4 plasmid types. DNA is replicated,
 +
transcribed (1) and proteins are synthesized (2). Capsid assembly
 +
occurs (3) and single-stranded DNA is packaged into the viral particle
 +
(4).<br>
 +
</div>
</td>
</td>
-
<td>
+
<td> <img
-
<center>
+
src="https://static.igem.org/mediawiki/2010/b/b8/Freiburg10_VirusProduction01.png"
-
<img width="477" src="https://static.igem.org/mediawiki/2010/b/b8/Freiburg10_VirusProduction01.png" alt="Reaction scheme for the virus production" />
+
alt="Reaction scheme for the virus production" width="477"><br>
-
</center>
+
<br>
</td>
</td>
 +
</tr>
 +
</tbody>
</table>
</table>
-
<br><br>
+
<br>
 +
<br>
<h2>Reduced Reaction Scheme</h2>
<h2>Reduced Reaction Scheme</h2>
-
Even the coarse model for virus production described in the previous paragraph would still consist of 24 ODEs containing 39 parameters (35 rate constants and 4 initial plasmid concentrations). Taking into account the linearity of the law of mass action (LMA) for simple transport processes we can neglect these fast reactions and for this reason reduce the model to the rate limiting steps like protein synthetization, capsid formation and virus packaging.<br>
+
<p style="text-align: justify;">
-
<br><br>
+
Even the coarse model for virus production described in the previous
-
 
+
paragraph would still consist of 24 ODEs containing 39 parameters (35
 +
rate constants and 4 initial plasmid concentrations). Taking into
 +
account the linearity of the law of mass action (LMA) for simple
 +
transport processes we can neglect these fast reactions and therefore reduce the model to the rate limiting steps like protein
 +
synthetization, capsid formation and virus packaging.<br>
 +
</p>
 +
<br>
 +
<br>
<center>
<center>
-
<img width="636" height="558" src="https://static.igem.org/mediawiki/2010/6/68/Freiburg10_VirusProduction02.png" alt="reduced reaction scheme for the virus production" />
+
<img
-
</center>
+
src="https://static.igem.org/mediawiki/2010/6/68/Freiburg10_VirusProduction02.png"
-
 
+
alt="reduced reaction scheme for the virus production" width="503"></center>
-
<br><br>
+
<br>
 +
<br>
<h2>Differential Equations</h2>
<h2>Differential Equations</h2>
-
The 13 reactions for the virus production are represented in a system of 17 coupled ODEs.<br>
+
<p style="text-align: justify;">
-
In addition to the terms provided by the law of mass action we considered the following terms:<br>
+
The 13 reactions for the virus production are represented in a system
 +
of 17 coupled ODEs.<br>
 +
In addition to the terms provided by the law of mass action we
 +
considered the following terms:<br>
 +
</p>
<ul>
<ul>
-
<li> a linear degradation of <i>ssDNA</i> in the nucleus with the rate constant <i>k<sub>14,1</sub></i>
+
<li> a linear degradation of <i>ssDNA</i> in the nucleus with the
-
<li> replication of <i>ssDNA</i> in the nucleus with the rate constant <i>k<sub>15,1</sub></i>
+
rate constant <i>k<sub>14,1</sub></i> </li>
 +
<li> replication of <i>ssDNA</i> in the nucleus with the rate
 +
constant <i>k<sub>15,1</sub></i> </li>
</ul>
</ul>
-
<br><br>
+
<br>
-
 
+
<br>
<center>
<center>
-
<img width="640" height="779" src="https://static.igem.org/mediawiki/2010/2/24/Freiburg10_VirusProduction03.png" alt="Reaction scheme for the virus production" />
+
<img
-
</center>
+
src="https://static.igem.org/mediawiki/2010/2/24/Freiburg10_VirusProduction03.png"
-
<br><br>
+
alt="Reaction scheme for virus production" width="701"></center>
 +
<br>
 +
<br>
<h2>Methods and Simulation</h2>
<h2>Methods and Simulation</h2>
-
The ODE model was implemented in MathWorks® MATLAB R2010b. Integration of the differential equations was achieved using the stiff integrator <i>ode15s</i> with automatic integration step size management.<br>
+
<p style="text-align: justify;">
-
To calibrate the dynamics of the mathematical model to those of biological system we used time lapse data of fluorescence experiments as well as published values for the rate constants.
+
The ODE model was implemented in MathWorks® MATLAB R2010b. Integration
-
<br><br>
+
of the differential equations was achieved using the stiff integrator <i>ode15s</i>
 +
with automatic integration step size management.<br>
 +
In order to adjust the dynamical model to biological data we extracted
 +
the average intensity out of the time lapse recordings of fluorescence
 +
experiments as well as published values for the rate constants. For
 +
initial conditions we took the plasmid concentrations we used in
 +
experiments.<br>
 +
<br>
 +
</p>
 +
<table
 +
style="text-align: left; width: 905px; margin-left: auto; margin-right: auto;"
 +
border="0" cellpadding="2" cellspacing="2">
 +
<tbody>
 +
<tr>
 +
<td style="vertical-align: top; width: 350px;"><a
 +
style="font-weight: bold;"
 +
href="https://static.igem.org/mediawiki/2010/b/b1/Freiburg10_VirusProductionTimeLapsemVenus.gif"><img
 +
src="https://static.igem.org/mediawiki/2010/d/d7/Freiburg10_VirusProductionTimeLapsemVenus.png"
 +
alt="" width="350"></a><br style="font-weight: bold;">
 +
<div style="text-align: justify;"><span style="font-weight: bold;">Figure
 +
2: </span>Fluorescence microscopy of transfected cells.&nbsp; mVenus
 +
is included to the modified capsid plasmid i.e. fluorescence intensity
 +
reflects capsid protein concentration.<br>
 +
</div>
 +
</td>
 +
<td>
 +
<p style="text-align: justify;">The image on the left shows one
 +
snapshot out of the time
 +
lapse recorded over a period of 1560 minutes (26 hours) after
 +
transfection. The bright spots
 +
correspond to the fluorescence intensity of <i>mVenus</i> in the upper
 +
and of <i>mCherry</i> in the lower picture.</p>
 +
<p style="text-align: justify;">To see the whole time lapse as an
 +
animation just click on the picture!<br>
 +
</p>
 +
</td>
 +
</tr>
 +
<tr>
 +
<td style="vertical-align: top; width: 350px;"><a
 +
href="https://static.igem.org/mediawiki/2010/1/1b/Freiburg10_VirusProductionTimeLapsemCherry.gif"><img
 +
src="https://static.igem.org/mediawiki/2010/e/e5/Freiburg10_VirusProductionTimeLapsemCherry.png"
 +
alt="" width="350"></a><br style="font-weight: bold;">
 +
<div style="text-align: justify;"><span style="font-weight: bold;">Figure
 +
3:</span> <span style="font-weight: bold;"></span>Fluorescence
 +
microscopy of
 +
transfected cells.&nbsp; Viral particles containing mCherry as gene of
 +
interest are visible. </div>
 +
</td>
 +
<td style="vertical-align: top;"><br>
 +
</td>
 +
</tr>
 +
</tbody>
 +
</table>
 +
<br>
 +
<center>
 +
<table style="text-align: left; width: 905px; height: 781px;" border="0"
 +
cellpadding="2" cellspacing="2">
 +
<tbody>
 +
<tr>
 +
<td style="vertical-align: top; width: 597px;"><img
 +
src="https://static.igem.org/mediawiki/2010/d/d5/Freiburg10_VirusProductionData01.png"
 +
alt="Data"><br>
 +
<div style="text-align: justify;"><span style="font-weight: bold;">Figure
 +
4: A</span> shows the average intensity of mCherry recorded using
 +
fluorescence microscopy. The curve corresponds to the rising phase of
 +
protein concentration and is expected to saturate for longer times as
 +
the harvest of viral particles is done after 3 days (4320min). <span
 +
style="font-weight: bold;">B: </span>time course of the intensity of
 +
mCherry. Due to the weak expression of mCherry the signal to noise
 +
ratio is quiet low and the functional dependency is not clearly
 +
determinable.<br>
 +
</div>
 +
</td>
 +
<td>
 +
<p style="text-align: justify;">The average intensity was
 +
extracted from the raw data uisng a script written in MathWorks®
 +
MATLAB which sums up the intensity value of each pixel of one image. <br>
 +
</p>
 +
<a
 +
href="https://static.igem.org/mediawiki/2010/1/19/Freiburg10_IntensityAnalysisCode.m">Download
 +
the m-File (MATLAB source code).</a> <br>
 +
</td>
 +
</tr>
 +
</tbody>
 +
</table>
 +
</center>
 +
<br>
The used model parameters are given in the table below.
The used model parameters are given in the table below.
<br>
<br>
-
 
-
<img width="405" height="337" src="https://static.igem.org/mediawiki/2010/4/49/Freiburg10_RateConstants01.png" alt="" />
 
<br>
<br>
-
 
+
<table
-
 
+
style="text-align: left; width: 300px; margin-left: auto; margin-right: auto;"
-
<a href="https://static.igem.org/mediawiki/2010/8/87/Freiburg10_VirusProductionCode.m">Download the m-File (MATLAB source code).</a><br>  
+
border="0" cellpadding="2" cellspacing="2">
-
<br><br>
+
<tbody>
 +
<tr>
 +
<td style="vertical-align: top;">
 +
<div style="text-align: center;"><img
 +
src="https://static.igem.org/mediawiki/2010/4/49/Freiburg10_RateConstants01.png"
 +
alt="" width="405"><br style="font-weight: bold;">
 +
</div>
 +
<div style="text-align: justify;"><span style="font-weight: bold;"><br>
 +
Table 1: </span>Rate constants for the virus production model.
 +
Generally forward reactions were assumed to be faster than reverse
 +
ones.
 +
Replication of <span style="font-style: italic;">ssDNA</span> is
 +
slower than its degradation.<br>
 +
</div>
 +
</td>
 +
</tr>
 +
</tbody>
 +
</table>
 +
<br>
 +
<br>
 +
&nbsp;
 +
<br>
 +
<a
 +
href="https://static.igem.org/mediawiki/2010/8/87/Freiburg10_VirusProductionCode.m">Download
 +
the m-File (MATLAB source code).</a><br>
 +
<br>
 +
<br>
<h2>Results and Discussion</h2>
<h2>Results and Discussion</h2>
 +
<p style="text-align: justify;">
 +
Figure 5 shows the time course of the model for virus production. The
 +
initial plasmid concentrations were chosen to 20µM for the <i>helper</i>,
 +
<i>goi</i>, and <i>capsid-wt</i> plasmids and 10µM of the modified
 +
capsid plasmid. After the short peaks of the intranuclear plamid
 +
concentrations proteins are synthesized and capsids are formed. The
 +
ssDNA enters the capsid through a pore and infectious virus particles
 +
are reseased to the cytoplasm from where they are transported out of
 +
the cell.<br>
 +
The concentrations reach a steady state as a result of ssDNA
 +
degradation inside the nucleus.<br>
 +
</p>
<center>
<center>
-
<img width=800 src="https://static.igem.org/mediawiki/2010/3/32/Freiburg10_VirusProductionPlot01.png" alt="" />
+
<table
 +
style="text-align: left; width: 642px; height: 504px; margin-left: 0px; margin-right: 0px;"
 +
border="0" cellpadding="2" cellspacing="2">
 +
<tbody>
 +
<tr>
 +
<td style="vertical-align: top;"><img
 +
src="https://static.igem.org/mediawiki/2010/3/32/Freiburg10_VirusProductionPlot01.png"
 +
alt="" width="800"><br>
 +
<div style="text-align: justify;"><span style="font-weight: bold;">Figure
 +
5:</span> <br>
 +
</div>
 +
</td>
 +
</tr>
 +
</tbody>
 +
</table>
 +
<br>
 +
<p style="text-align: justify;">A model extension was made taking into
 +
account the production efficiency dependend on the level of
 +
modfication. The resulting curves are plotted in figure 6. <br>
 +
</p>
</center>
</center>
-
<br><br>
+
<br>
-
<center>
+
<br>
-
<img width=800 src="https://static.igem.org/mediawiki/2010/a/ac/Freiburg10_VirusProductionPlot02.png" alt="" />
+
<table
-
</center>
+
style="text-align: left; width: 758px; height: 773px; margin-left: auto; margin-right: auto;"
-
 
+
border="0" cellpadding="2" cellspacing="2">
-
<img width=800 src="https://static.igem.org/mediawiki/2010/d/db/Freiburg10_VirusProductionModPlot01.png" alt="" />
+
<tbody>
-
 
+
<tr>
-
<br><br>
+
<td style="vertical-align: top;">
 +
<div style="text-align: center;"><img alt=""
 +
src="https://static.igem.org/mediawiki/2010/d/db/Freiburg10_VirusProductionModPlot01.png"
 +
width="800"><br>
 +
</div>
 +
<div style="text-align: justify;"> <span
 +
style="font-weight: bold;">Figure
 +
6: A:</span> Enzyme concentration depending on different degrees of
 +
modification. <span style="font-weight: bold;">B:</span> production
 +
efficiency as a function of modification degree <span
 +
style="font-style: italic;">m</span>.<br>
 +
</div>
 +
</td>
 +
</tr>
 +
</tbody>
 +
</table>
 +
<br>
 +
<p style="text-align: justify;">
 +
Fitting the model to the data obtained from the fluorescence experiment
 +
was performed by the method of least squares in logarrithmic parameter
 +
space. Unfortunately only one data set quantifying the single protein,
 +
capsid and virus concentration was availible so that the optimization
 +
problem was clearly under-determined and no explicit ideal parameter
 +
set could be found without loosing the biological signification. More
 +
precisely, the data was fitted by an exponential increase while the
 +
biological system is expected to saturate for large time values because
 +
no more virus is poduced if its plasmid are completely degraded. Figure
 +
7 shows more realistic model characteristics. The blue line
 +
represents the sum of all concentrations containing mVenus and the red
 +
dots describe the mVenus intensity data set. Thereby the desired
 +
simoidal shape was achieved though the chi-square value and
 +
consequently the quality of the fit is not optimal. <br>
 +
In order to improve the predictive capability of this mathematical
 +
model one has to perform further adjustments and more experimental data
 +
is needed accordingly.<br>
 +
</p>
 +
<br>
 +
<table
 +
style="text-align: left; width: 683px; height: 663px; margin-left: auto; margin-right: auto;"
 +
border="0" cellpadding="2" cellspacing="2">
 +
<tbody>
 +
<tr>
 +
<td style="vertical-align: top;"><span style="font-weight: bold;"><img
 +
style="width: 800px; height: 599px;" alt=""
 +
src="https://static.igem.org/mediawiki/2010/7/74/Freiburg10_VirusProductionFit01.png"><br>
 +
</span>
 +
<div style="text-align: justify;"><span style="font-weight: bold;">Figure
 +
7:</span> Data fitting approach. The model can be fitted perfectly to
 +
the data (not shown) but is not meaningful in a biological sense
 +
because exponential increase does not occur in this system. Considering
 +
the fact that virus concentration should saturate a sigmoidal shape is
 +
expected. Such a fit was not achieved because to many unknown
 +
parameters for one single data set.<br>
 +
</div>
 +
</td>
 +
</tr>
 +
</tbody>
 +
</table>
 +
<br>
 +
<br>
 +
<br>
</html>
</html>
{{:Team:Freiburg_Bioware/Footer}}
{{:Team:Freiburg_Bioware/Footer}}

Latest revision as of 02:01, 28 October 2010

Model for Virus Production


Reaction Scheme

Reducing the complexity of virus production we divide the cell into three compartments: the extracellular matrix (all quantities with the index ext), the cytoplasm (cyt) and the nucleus (nuc). Four plasmids are transfected - the plasmid coding for the helper proteins (helper), the gene of interest (goi) and two types of plasmids coding for the capsid proteins (capwt [wild type], capmod [modified]).
The plasmids are transported into the nucleus where gene expression is initiated. Processed mRNA is transported into the cytoplasm and proteins (phelper, pcapwt, pcapmod) are produced. Containing a nuclear localization sequence proteins are relocated into the nucleus where capsid assembly occurs. The viral capsid is composed of 60 subunits of viral coat proteins. Titration of the two plasmids coding for the capsid proteins leads to virus surfaces with different ratios of wild type and modified capsid proteins.
The gene of interest is replicated by cellular polymerases and single stranded DNA (ssDNA) is encapsidated into the preformed capsids (capsid) forming infectious viral particles (V).
Finally the recombinant viruses are released into the extracellular matrix and can be harvested for transduction.



Reaction scheme for the virus production



Figure 1: Schematic overview of virus production: A production cell line is  transfected with 4 plasmid types. DNA is replicated, transcribed (1) and proteins are synthesized (2). Capsid assembly occurs (3) and single-stranded DNA is packaged into the viral particle (4).
Reaction scheme for the virus production



Reduced Reaction Scheme

Even the coarse model for virus production described in the previous paragraph would still consist of 24 ODEs containing 39 parameters (35 rate constants and 4 initial plasmid concentrations). Taking into account the linearity of the law of mass action (LMA) for simple transport processes we can neglect these fast reactions and therefore reduce the model to the rate limiting steps like protein synthetization, capsid formation and virus packaging.



reduced reaction scheme for the virus production


Differential Equations

The 13 reactions for the virus production are represented in a system of 17 coupled ODEs.
In addition to the terms provided by the law of mass action we considered the following terms:

  • a linear degradation of ssDNA in the nucleus with the rate constant k14,1
  • replication of ssDNA in the nucleus with the rate constant k15,1


Reaction scheme for virus production


Methods and Simulation

The ODE model was implemented in MathWorks® MATLAB R2010b. Integration of the differential equations was achieved using the stiff integrator ode15s with automatic integration step size management.
In order to adjust the dynamical model to biological data we extracted the average intensity out of the time lapse recordings of fluorescence experiments as well as published values for the rate constants. For initial conditions we took the plasmid concentrations we used in experiments.


Figure 2: Fluorescence microscopy of transfected cells.  mVenus is included to the modified capsid plasmid i.e. fluorescence intensity reflects capsid protein concentration.

The image on the left shows one snapshot out of the time lapse recorded over a period of 1560 minutes (26 hours) after transfection. The bright spots correspond to the fluorescence intensity of mVenus in the upper and of mCherry in the lower picture.

To see the whole time lapse as an animation just click on the picture!


Figure 3: Fluorescence microscopy of transfected cells.  Viral particles containing mCherry as gene of interest are visible.


Data
Figure 4: A shows the average intensity of mCherry recorded using fluorescence microscopy. The curve corresponds to the rising phase of protein concentration and is expected to saturate for longer times as the harvest of viral particles is done after 3 days (4320min). B: time course of the intensity of mCherry. Due to the weak expression of mCherry the signal to noise ratio is quiet low and the functional dependency is not clearly determinable.

The average intensity was extracted from the raw data uisng a script written in MathWorks® MATLAB which sums up the intensity value of each pixel of one image.

Download the m-File (MATLAB source code).

The used model parameters are given in the table below.



Table 1:
Rate constants for the virus production model. Generally forward reactions were assumed to be faster than reverse ones. Replication of ssDNA is slower than its degradation.


 
Download the m-File (MATLAB source code).


Results and Discussion

Figure 5 shows the time course of the model for virus production. The initial plasmid concentrations were chosen to 20µM for the helper, goi, and capsid-wt plasmids and 10µM of the modified capsid plasmid. After the short peaks of the intranuclear plamid concentrations proteins are synthesized and capsids are formed. The ssDNA enters the capsid through a pore and infectious virus particles are reseased to the cytoplasm from where they are transported out of the cell.
The concentrations reach a steady state as a result of ssDNA degradation inside the nucleus.


Figure 5:

A model extension was made taking into account the production efficiency dependend on the level of modfication. The resulting curves are plotted in figure 6.




Figure 6: A: Enzyme concentration depending on different degrees of modification. B: production efficiency as a function of modification degree m.

Fitting the model to the data obtained from the fluorescence experiment was performed by the method of least squares in logarrithmic parameter space. Unfortunately only one data set quantifying the single protein, capsid and virus concentration was availible so that the optimization problem was clearly under-determined and no explicit ideal parameter set could be found without loosing the biological signification. More precisely, the data was fitted by an exponential increase while the biological system is expected to saturate for large time values because no more virus is poduced if its plasmid are completely degraded. Figure 7 shows more realistic model characteristics. The blue line represents the sum of all concentrations containing mVenus and the red dots describe the mVenus intensity data set. Thereby the desired simoidal shape was achieved though the chi-square value and consequently the quality of the fit is not optimal.
In order to improve the predictive capability of this mathematical model one has to perform further adjustments and more experimental data is needed accordingly.



Figure 7: Data fitting approach. The model can be fitted perfectly to the data (not shown) but is not meaningful in a biological sense because exponential increase does not occur in this system. Considering the fact that virus concentration should saturate a sigmoidal shape is expected. Such a fit was not achieved because to many unknown parameters for one single data set.